Download - The Natural Logarithmic Function
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The Natural Logarithmic
FunctionDifferentiation
Integration
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Properties of the Natural Log Function
• If a and b are positive numbers and n is rational, then the following properties are true: ln(1) 0
ln( ) ln lnab a b
ln( ) lnna n a
ln ln lna a bb
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The Algebra of Logarithmic Expressions
10ln ln10 ln 99
1/ 2 1ln 3 2 ln 3 2 ln 3 22
x x x
6ln ln 6 ln 55x x
22
3 2
3ln
1
x
x x
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The Derivative of the Natural Logarithmic Function
Let u be a differentiable function of x
1ln , 0d x xdx x
ln , 0d uu udu u
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Differentiation of Logarithmic Functions
ln 2d xdx
2ln 1d xdx
lnd x xdx
3lnd xdx
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Differentiation of Logarithmic Functions
ln 2d xdx
2 2u x u
ln , 0d uu udu u
2 12
uu x x
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Differentiation of Logarithmic Functions
2ln 1d xdx
2 1 2u x u x
ln , 0d uu udu u
2
21
u xu x
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Differentiation of Logarithmic Functions
ln APPLY PRODUCT RULE!!!!!!!!!!d x xdx
11 ln x xx
ln 1x
ln ln lnd d dx x x x x xdx dx dx
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Differentiation of Logarithmic Functions
3ln CHAIN RULE!!!d xdx
23 ln lndx xdx
23 ln xx
2 13 ln xx
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Logarithmic Properties as Aids to Differentiation
• Differentiate: ln 1f x x
1/2 1ln 1 ln 12
f x x x
11 12 1 2 1x
f xx x
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Logarithmic Properties as Aids to Differentiation
• Differentiate:
22
3
1ln
2 1
x xf x
x
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Logarithmic Differentiation
• Differentiate:
This can get messy with the quotient or product and chain rules. So we will use ln rules to help simplify this and apply implicit differentiation and then we solve for y’…
2
2
2
1
xf x
x
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Derivative Involving Absolute Value
• Recall that the ln function is undefined for negative numbers, so we often see expressions of the form ln|u|. So the following theorem states that we can differentiate functions of the form y= ln|u| as if the absolute value symbol is not even there.
• If u is a differentiable function such that u≠0 then:
lnd uudu u
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Derivative Involving Absolute Value
• Differentiate: ln cosf x x
ln cosd uxdx u
cos , sinu u x u xu
sincos
u xu x
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Finding Relative Extrema
• Locate the relative extrema of • Differentiate:
• Set = 0 to find critical points=02x+2=0X=-1, Plug back into original to find yy=ln(1-2+3)=ln2 So, relative extrema is at (-1, ln2)
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Homework• 5.1 Natural Logarithmic Functions and the
Number e Derivative #19-35,47-65, 71,79,93-96
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General Power Rule for Integration
• Recall that it has an important disclaimer- it doesn’t apply when n = -1. So we can not integrate functions such as f(x)=1/x.
• So we use the Second FTC to DEFINE such a function.
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Integration Formulas• Let u be a differentiable function of x
1 lndx x cx
1 lndx u cu
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Using the Log Rule for Integration
2dxx
1Factor out the constant:2
This gives us a form we recognize and can easily integrate.
dxx
2ln x c
2Use rules to clean things up:ln x c
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Using the Log Rule with a Change of Variables
( 14 𝑥−1 )𝑑𝑥 Let u=4x-1, so du=4dx
and dx=1 ln4
u c
1 ln 4 14
x c
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Finding Area with the Log Rule
• Find the area of the region bounded by the graph of y, the x-axis and the line x=3.
3 3
20 0Set up your integral: ( )
1xy x dx dx
x
We have an integral in the form u
u
2 1 so 2Let u x u xdx
2
1 1 1 122 1 2
xdx dux u
32
0
1 1ln ln 12 2
u x
2 21 1 1 1 1ln 3 1 ln 0 1 ln10 ln 1 ln10 ln 102 2 2 2 2
2 1xy x
x
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Recognizing Quotient Forms of the Log Rule
23 3
3
3 1 lnx dx u x x x x cx x
2sec tan ln tantan
x dx u x x cx
1/31 1 3 13 2 ln 3 2 ln 3 23 2 3 3 2 3
dx u x dx x c x cx x
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DefinitionThe natural logarithmic function is defined
by
The domain of the natural logarithmic function is the set of all positive real numbers
1
1ln , 0x
x dt xt
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u-Substitution and the Log Rule1 _ _ln
dy a differential equationdx x x
1_ _ :ln
Integrate both sides y dxx x
1lnu x du dxx
u duu
ln u c
ln ln x c
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Long Division With Integrals
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How you know it’s long Division
• If it is top heavy that means it is long division.o Example
3 2
2
4 4 96 10025
x x xx
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Example 12 5 6
5x xx
25 5 6x x x
2
2
65
5 5 6
-x 5
xx
x x x
x
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Continue Example 1
21 6ln 52x x
65
xx
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Example 22 3 2
1x xx
21 3 2x x x
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Continue Example 22
2
21 3 2
-x 1 2 2 -2x -2
xx x x
xx
2x 21 22x x
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Using Long Division Before Integrating
2
2
11
x x dxx
22 2
2 2
11 1 1 1
1 1
rxx x xdx x x xx x
211
x dxx
2 1xdx dx
x
21 ln 12
x x c
2: 1 0, ... _ _ _ _ _Note x x so no need for absolute value
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Using a Trigonometric Identity
tan xdx
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Guidelines for integration
1. Learn a basic list of integration formulas. (including those given in this section, you now have 12 formulas: the Power Rule, the Log Rule, and ten trigonometric rules. By the end of section 5.7 , this list will have expanded to 20 basic rules)
2. Find an integration formula that resembles all or part of the integrand, and, by trial and error, find a choice of u that will make the integrand conform to the formula.
3. If you cannot find a u-substitution that works, try altering the integrand. You might try a trigonometric identity, multiplication and division by the same quantity, or addition and subtraction of the same quantity. Be creative.
4. If you have access to computer software that will find antiderivatives symbolically, use it.
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Integrals of the Six Basic Trigonometric Functions
sin cosudu u c cos sinudu u c tan ln cosudu u c cot ln sinudu u c
sec ln sec tanudu u u c csc ln csc cotudu u u c
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Homework• 5.2 Log Rule for Integration and Integrals for Trig
Functions (substitution) #1-39, 47-53, 67